Math Problem Statement

Answer this question

Find the derivative of the following functions: a) y = tan⁻¹(√(1 + x² - 1) / x) b) y = (x² + 2x + 1) / √(x + 1) c) y = (x³√(x + 2)) / ((x + 2)(x - 3)) d) y = x²(x - 2)e^(sin x)

Solution

Let's go over each part of the problem one by one:

a) y=tan1(1+x21x)y = \tan^{-1} \left( \frac{\sqrt{1+x^2-1}}{x} \right)

We can rewrite this as:

y=tan1(xx)y = \tan^{-1} \left( \frac{x}{x} \right)

Which simplifies to:

y=tan1(1)y = \tan^{-1} (1)

Since tan1(1)=π4\tan^{-1}(1) = \frac{\pi}{4}, the derivative is 00 because the function is a constant.

Derivative: dydx=0\frac{dy}{dx} = 0

b) y=x2+2x+1x+1y = \frac{x^2 + 2x + 1}{\sqrt{x+1}}

We will apply the quotient rule for derivatives:

dydx=(2x+2)x+1(x2+2x+1)12x+1(x+1)2\frac{dy}{dx} = \frac{ \left( 2x + 2 \right) \sqrt{x+1} - \left( x^2 + 2x + 1 \right) \cdot \frac{1}{2\sqrt{x+1}} }{ (\sqrt{x+1})^2 }

Simplify:

dydx=(2x+2)x+1(x2+2x+1)2x+1x+1\frac{dy}{dx} = \frac{ (2x+2) \sqrt{x+1} - \frac{(x^2 + 2x + 1)}{2 \sqrt{x+1}} }{ x+1 }

c) y=x3x+2(x+2)(x3)y = \frac{x^3 \sqrt{x+2}}{(x+2)(x-3)}

We can apply the quotient rule again here:

dydx=(3x2x+2+x312x+21)(x+2)(x3)x3x+2((x+2)+(x3))(x+2)2(x3)2\frac{dy}{dx} = \frac{ \left( 3x^2 \sqrt{x+2} + x^3 \cdot \frac{1}{2\sqrt{x+2}} \cdot 1 \right)(x+2)(x-3) - x^3 \sqrt{x+2} \left( (x+2) + (x-3) \right) }{ (x+2)^2 (x-3)^2 }

d) y=x2(x2)esinxy = x^2 (x-2) e^{\sin x}

To differentiate this, we apply the product rule:

dydx=(2x(x2)+x2)esinx+x2(x2)esinxcosx\frac{dy}{dx} = \left( 2x (x-2) + x^2 \right) e^{\sin x} + x^2 (x-2) \cdot e^{\sin x} \cdot \cos x

This can be simplified further if needed.

Would you like further clarification on any of the steps or help with simplification?


5 Related Questions:

  1. How do you apply the quotient rule in more complex cases?
  2. What are the steps to differentiate trigonometric functions that include a composition like esinxe^{\sin x}?
  3. What is the general process for simplifying the result of a derivative calculation?
  4. Can you explain the product rule with an example involving more than two functions?
  5. How do you deal with nested functions in derivatives?

Tip: Always check for simplifications before proceeding with differentiation. Reducing the expression as much as possible can make the process smoother and avoid unnecessary complexity!

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Math Problem Analysis

Mathematical Concepts

Derivatives
Chain Rule
Product Rule
Quotient Rule
Inverse Trigonometric Functions

Formulas

Derivative of tan⁻¹(x) = 1 / (1 + x²)
Derivative of e^(sin x) = e^(sin x) * cos x
Quotient Rule: (f/g)' = (g*f' - f*g') / g²
Product Rule: (f * g)' = f' * g + f * g'

Theorems

Chain Rule
Quotient Rule
Product Rule

Suitable Grade Level

Grades 11-12